Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C33×Q16 ⋊ C22 |
| Sylow | C33 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×#1113 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C22×#1113 |
| Fitting | C33×Q16 | C22 |
| socle | C2×C33 | C22×D4 |
| hypercenter | Q16 | #1113 |
| hypocenter | C33 | #576 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
1276 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cf,d^a=df,g^a=g^2,h^a=h^2,c^b=cf,d^b=df,g^b=g^2,i^b=i^2,d^c=de,e^c=ef,e^d=ef99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44552);